How can I test the Blender 2.8 game engine on Linux?
As far as I know, there is no work being done on 2.8's game engine at the moment (Feb 2018). Eevee is a viewport rendering engine and has not made its way to the BGE yet.

As for UPBGE, it seems there is some work being done on getting Eevee shaders to work there back in last year, but stopped.